The Bay Area added 5,600 jobs during June, a sturdy performance that eases fears of an employment slowdown in the region. Meanwhile, the statewide jobless rate improved to its lowest level in nearly five years, labor officials reported Thursday.

Santa Clara County gained 3,500 jobs in June, the Alameda County-Contra Costa County area added 1,700 and the San Francisco-San Mateo-Marin region gained 1,200, the state’s Employment Development Department showed.

“The job growth is starting to spread beyond the South Bay and the San Francisco area,” said Jon Haveman, chief economist with Marin Consulting. “The East Bay is now part of the jobs rebound.”

California added 30,200 jobs in June, which means the Bay Area accounted for about one-fifth of the statewide gains.